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When positive integer n is divided by 3, the remainder is 2 [#permalink]

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When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

(1) n-2 is divisible by 5. (2) t is divisible by 3.

"If you want my advice, Peter," he said at last, "you've made a mistake already. By asking me. By asking anyone. Never ask people. Not about your work. Don't you know what you want? How can you stand it, not to know?" Ayn Rand

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

(1) n-2 is divisible by 5. (2) t is divisible by 3.

From the stem: \(n=3p+2\) and \(t=5q+3\). \(nt=15pq+9p+10q+6\), we should find the remainder when this expression is divided by 15.

(1) \(n-2=5m\) --> \(n=5m+2=3p+2\) --> \(5m=3p\), \(15m=9p\) --> \(nt=15pq+9p+10q+6=15pq+15m+10q+6\). Clearly \(15pq\) and \(15m\) are divisible by 15, so remainder by dividing these components will be 0. But we still know nothing about \(10q+6\). Not sufficient.

(2) t is divisible by 3 means that \(5q+3\) is divisible by 3 --> 5q is divisible by 3 or q is divisible by 3 --> \(5q=5*3z=15z\) --> \(10q=30z\) --> \(nt=15pq+9p+10q+6=15pq+9p+30z+6\). \(15pq\) and \(30z\) are divisible by 15. Know nothing about \(9p+6\). Not sufficient.

(1)+(2) \(9p=15m\) and \(10q=30z\) --> \(nt=15pq+9p+10q+6=15pq+15m+30z+6\). Remainder when this expression is divided by 15 is 6. Sufficient.

Answer: C.

OR:

From the stem: \(n=3p+2\) and \(t=5q+3\).

(1) n-2 is divisible by 5 --> \(n-2=5m\) --> \(n=5m+2\) and \(n=3p+2\) --> general formula for \(n\) would be \(n=15k+2\) (about deriving general formula for such problems at: good-problem-90442.html#p723049 and manhattan-remainder-problem-93752.html#p721341) --> \(nt=(15k+2)(5q+3)=15*5kq+15*3k+10q+6\) --> first two terms are divisible by 15 (\(15*5kq+15*3k\)) but we don't know about the last two terms (\(10q+6\)). Not sufficient.

(2) t is divisible by 3 --> \(t=3r\) and \(t=5q+3\) --> general formula for \(t\) would be \(t=15x+3\) --> \(nt=(3p+2)(15x+3)=15*3px+9p+15*2x+6\). Not sufficient.

(1)+(2) \(nt=(15k+2)(15x+3)=15*15kx+15*3k+15*2x+6\) this expression divided by 15 yields remainder of 6. Sufficient.

If the explanation above is not helpful, you may find a step by step video solution of this question useful. On GMATFix site, this is GMATPrep question 1045

Saw this question on a GMATPrep test, and I can't figure out how to get to the correct answer. Can anyone help? Thanks!

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

(1) n-2 is divisible by 5. (2) t is divisible by 3.

The correct answer is (C) - both statements together are sufficient, but neither statement alone is sufficient. Can anybody out there help explain to me how to get to this answer though? Thanks!

This is how I would approach this question.

When positive integer n is divided by 3, the remainder is 2; I say n = 3a + 2 ( a is a non negative integer)

and when positive integer t is divided by 5, the remainder is 3. So t = 5b + 3 (b is a non negative integer.)

What is the remainder when the product nt is divided by 15? So nt = (3a + 2)(5b + 3) = 15ab + 9a + 10b + 6 15ab is divisible by 15. But I don't know anything about (9a + 10b + 6) yet.

Stmnt 1: n-2 is divisible by 5. From above, n - 2 is just 3a. If n - 2 is divisible by 5, then 'a' must be divisible by 5. So I get that 9a is divisible by 15. I still don't know anything about b. If b = 1, remainder of nt is 1. If b = 2, remainder of nt is 11 and so on... Not sufficient.

Stmnt 2: t is divisible by 3. If t is divisible by 3, then (5b + 3) is divisible by 3. Therefore, b must be divisible by 3. (If this is unclear, think: 15 + 3 will be divisible by 3 but 20 + 3 will not be. If the second term is 3, the first term must also be divisible by 3 to make the whole expression divisible by 3). So 10b is divisible by 15 but we do not know anything about a. If a = 1, remainder of nt is 0, if a = 2, remainder of nt is 9. Not sufficient.

Using both statements together, we know 9a and 10b are divisible by 15. So remainder must be 6. Sufficient.

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

(1) n-2 is divisible by 5. (2) t is divisible by 3.

Answer: C.

I have another approach to this ds, plz correct me if i'm wrong. n=3x + 2 t = 5y + 3 Clearly we cannot solve the problem with either n or t. We need both information concerning n and t because we need to figure out the remaining of n*t. => left with C or E

(1): n-2 is divisible by 5 & n=3x + 2 => x is multiple of 5. (2): t is divisible by 3. & t = 5y + 3 => y is multiple of 3 (1)& (2) => n*t = (3x+2) (5y+3) = (3x*5y) + (9x) + (10y) + 6 we know that: x is multiple of 5, y is multiple of 3 so: (3x*5y) + (9x) + (10y) + 6 will have remaining of 6 because: each (3x*5y); (9x); (10y) is all multiple of 15. _________________

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Re: When positive integer n is divided by 3, the remainder is 2 [#permalink]

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07 Oct 2013, 21:15

Hello from the GMAT Club BumpBot!

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Saw this question on a GMATPrep test, and I can't figure out how to get to the correct answer. Can anyone help? Thanks!

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

(1) n-2 is divisible by 5. (2) t is divisible by 3.

The correct answer is (C) - both statements together are sufficient, but neither statement alone is sufficient. Can anybody out there help explain to me how to get to this answer though? Thanks!

This is how I would approach this question.

When positive integer n is divided by 3, the remainder is 2; I say n = 3a + 2 ( a is a non negative integer)

and when positive integer t is divided by 5, the remainder is 3. So t = 5b + 3 (b is a non negative integer.)

What is the remainder when the product nt is divided by 15? So nt = (3a + 2)(5b + 3) = 15ab + 9a + 10b + 6 15ab is divisible by 15. But I don't know anything about (9a + 10b + 6) yet.

Stmnt 1: n-2 is divisible by 5. From above, n - 2 is just 3a. If n - 2 is divisible by 5, then 'a' must be divisible by 5. So I get that 9a is divisible by 15. I still don't know anything about b. If b = 1, remainder of nt is 1. If b = 2, remainder of nt is 11 and so on... Not sufficient.

Stmnt 2: t is divisible by 3. If t is divisible by 3, then (5b + 3) is divisible by 3. Therefore, b must be divisible by 3. (If this is unclear, think: 15 + 3 will be divisible by 3 but 20 + 3 will not be. If the second term is 3, the first term must also be divisible by 3 to make the whole expression divisible by 3). So 10b is divisible by 15 but we do not know anything about a. If a = 1, remainder of nt is 0, if a = 2, remainder of nt is 9. Not sufficient.

Using both statements together, we know 9a and 10b are divisible by 15. So remainder must be 6. Sufficient.

Answer (C).

so from statement 1 we got 10b+6 if b=1 we get 16 then rem =1 if b=2 we got 106 so remainder =1 if b=3 we get 1006 so remainder =1 ..............................so i think a is sufficient .............what am i doing wrong

Saw this question on a GMATPrep test, and I can't figure out how to get to the correct answer. Can anyone help? Thanks!

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

(1) n-2 is divisible by 5. (2) t is divisible by 3.

The correct answer is (C) - both statements together are sufficient, but neither statement alone is sufficient. Can anybody out there help explain to me how to get to this answer though? Thanks!

This is how I would approach this question.

When positive integer n is divided by 3, the remainder is 2; I say n = 3a + 2 ( a is a non negative integer)

and when positive integer t is divided by 5, the remainder is 3. So t = 5b + 3 (b is a non negative integer.)

What is the remainder when the product nt is divided by 15? So nt = (3a + 2)(5b + 3) = 15ab + 9a + 10b + 6 15ab is divisible by 15. But I don't know anything about (9a + 10b + 6) yet.

Stmnt 1: n-2 is divisible by 5. From above, n - 2 is just 3a. If n - 2 is divisible by 5, then 'a' must be divisible by 5. So I get that 9a is divisible by 15. I still don't know anything about b. If b = 1, remainder of nt is 1. If b = 2, remainder of nt is 11 and so on... Not sufficient.

Stmnt 2: t is divisible by 3. If t is divisible by 3, then (5b + 3) is divisible by 3. Therefore, b must be divisible by 3. (If this is unclear, think: 15 + 3 will be divisible by 3 but 20 + 3 will not be. If the second term is 3, the first term must also be divisible by 3 to make the whole expression divisible by 3). So 10b is divisible by 15 but we do not know anything about a. If a = 1, remainder of nt is 0, if a = 2, remainder of nt is 9. Not sufficient.

Using both statements together, we know 9a and 10b are divisible by 15. So remainder must be 6. Sufficient.

Answer (C).

so from statement 1 we got 10b+6 if b=1 we get 16 then rem =1 if b=2 we got 106 so remainder =1 if b=3 we get 1006 so remainder =1 ..............................so i think a is sufficient .............what am i doing wrong

10b above means 10*b, 10 multiplied by b.

If b=2, then 10b+6=10*2+6=26 not 106; If b=2, then 10b+6=10*3+6=36 not 1006.

Re: When positive integer n is divided by 3, the remainder is 2 [#permalink]

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27 Oct 2014, 23:58

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: When positive integer n is divided by 3, the remainder is 2 [#permalink]

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11 Nov 2015, 20:57

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

n = 3a + 2, t = 5b + 3. Q: (3a + 2)(5b + 3)/15 = c + r/15. What is the value of r?

r is knowable only if it can be determined that each of 3a and 5b are divisible by 15, or in other words that a is divisible by 5 and b is divisible by 3, in which case the remainder would be the only remaining element that is clearly not divisible by 15.

nt = (3a +2)(5b +3) = 3a x 5b + 3a x3 + 10b + 6

if the first three elements of this equation are divisible by 15 the remainder would be 6. If divisibility of any of the variable-containing elements of this equation cannot be determined then the remainder cannot be determined.

St1: n-2 is divisible by 5.

n = 3a + 2 3a = n-2

if n-2 divisible by 5, then 3a is divisible by 5 (at least one factor of a = 5), and therefore 3a is divisible by 15.

However, divisibility of 5b by 15 cannot be determined, and therefore the value of r cannot be determined.

INSUFF.

(2) t is divisible by 3.

t = 5b + 3 5b + 3 is divisible by 3, therefore 5b is divisible by 3(at least one factor of b is 3) and 5b is divisible by 15.

However, divisibility of 3a by 15 cannot be determined, and therefore the value of r cannot be determined.

INSUFF.

Combined: 3a, and 5b are both divisible by 15, and therefore, all variable-containing elements of the equation nt = (3a +2)(5b +3) = 3a x 5b + 3a x3 + 10b + 6 are divisible by 15 and the remainder is 6.

Re: When positive integer n is divided by 3, the remainder is 2 [#permalink]

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28 Feb 2016, 10:09

Hi Bunuel,

thanks for the solution. I tried to solve statement two and I below is what I am getting. What am i doing wrong?

T = 5q+3 (given) T = 3k (statement 2) So 5q + 3 = 3k --> 10q + 6 = 6k and this does not assures that 10q+6 is a multiple of 15. What am i missing in solving it this way?

Thanks in advance!

Bunuel wrote:

When positive integer n is divided by 3, the remainder is 2; and when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product nt is divided by 15?

(1) n-2 is divisible by 5. (2) t is divisible by 3.

From the stem: \(n=3p+2\) and \(t=5q+3\). \(nt=15pq+9p+10q+6\), we should find the remainder when this expression is divided by 15.

(1) \(n-2=5m\) --> \(n=5m+2=3p+2\) --> \(5m=3p\), \(15m=9p\) --> \(nt=15pq+9p+10q+6=15pq+15m+10q+6\). Clearly \(15pq\) and \(15m\) are divisible by 15, so remainder by dividing these components will be 0. But we still know nothing about \(10q+6\). Not sufficient.

(2) t is divisible by 3 means that \(5q+3\) is divisible by 3 --> 5q is divisible by 3 or q is divisible by 3 --> \(5q=5*3z=15z\) --> \(10q=30z\) --> \(nt=15pq+9p+10q+6=15pq+9p+30z+6\). \(15pq\) and \(30z\) are divisible by 15. Know nothing about \(9p+6\). Not sufficient.

(1)+(2) \(9p=15m\) and \(10q=30z\) --> \(nt=15pq+9p+10q+6=15pq+15m+30z+6\). Remainder when this expression is divided by 15 is 6. Sufficient.

Answer: C.

OR:

From the stem: \(n=3p+2\) and \(t=5q+3\).

(1) n-2 is divisible by 5 --> \(n-2=5m\) --> \(n=5m+2\) and \(n=3p+2\) --> general formula for \(n\) would be \(n=15k+2\) (about deriving general formula for such problems at: good-problem-90442.html#p723049 and manhattan-remainder-problem-93752.html#p721341) --> \(nt=(15k+2)(5q+3)=15*5kq+15*3k+10q+6\) --> first two terms are divisible by 15 (\(15*5kq+15*3k\)) but we don't know about the last two terms (\(10q+6\)). Not sufficient.

(2) t is divisible by 3 --> \(t=3r\) and \(t=5q+3\) --> general formula for \(t\) would be \(t=15x+3\) --> \(nt=(3p+2)(15x+3)=15*3px+9p+15*2x+6\). Not sufficient.

(1)+(2) \(nt=(15k+2)(15x+3)=15*15kx+15*3k+15*2x+6\) this expression divided by 15 yields remainder of 6. Sufficient.

Answer: C.

gmatclubot

Re: When positive integer n is divided by 3, the remainder is 2
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28 Feb 2016, 10:09

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